## Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians

HTML articles powered by AMS MathViewer

- by Aaron Bertram, Georgios Daskalopoulos and Richard Wentworth PDF
- J. Amer. Math. Soc.
**9**(1996), 529-571 Request permission

## Abstract:

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as “Gromov invariants”) on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one.## References

- M. F. Atiyah and R. Bott,
*The Yang-Mills equations over Riemann surfaces*, Philos. Trans. Roy. Soc. London Ser. A**308**(1983), no. 1505, 523–615. MR**702806**, DOI 10.1098/rsta.1983.0017 - E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris,
*Geometry of algebraic curves. Vol. I*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR**770932**, DOI 10.1007/978-1-4757-5323-3 - Steven B. Bradlow,
*Special metrics and stability for holomorphic bundles with global sections*, J. Differential Geom.**33**(1991), no. 1, 169–213. MR**1085139** - Steven B. Bradlow and Georgios D. Daskalopoulos,
*Moduli of stable pairs for holomorphic bundles over Riemann surfaces*, Internat. J. Math.**2**(1991), no. 5, 477–513. MR**1124279**, DOI 10.1142/S0129167X91000272 - Bradlow, S. B. and G. D. Daskalopoulos,
*Moduli of stable pairs for holomorphic bundles over Riemann surfaces II*, Int. J. Math.**4**(No. 6) (1993), 903–925. - Bradlow, S., G. Daskalopoulos, and R. Wentworth,
*Birational equivalences of vortex moduli*, Topology (to appear). - Raoul Bott and Loring W. Tu,
*Differential forms in algebraic topology*, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR**658304**, DOI 10.1007/978-1-4757-3951-0 - Aaron Bertram,
*Moduli of rank-$2$ vector bundles, theta divisors, and the geometry of curves in projective space*, J. Differential Geom.**35**(1992), no. 2, 429–469. MR**1158344** - A. Bruguières,
*The scheme of morphisms from an elliptic curve to a Grassmannian*, Compositio Math.**63**(1987), no. 1, 15–40. MR**906377** - Daskalopoulos, G. and K. Uhlenbeck,
*An application of transversality to the topology of the moduli space of stable bundles*, Topology**34**(1994), 203–215. - Andreas Floer,
*Symplectic fixed points and holomorphic spheres*, Comm. Math. Phys.**120**(1989), no. 4, 575–611. MR**987770**, DOI 10.1007/BF01260388 - William Fulton,
*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR**732620**, DOI 10.1007/978-3-662-02421-8 - M. Gromov,
*Pseudo holomorphic curves in symplectic manifolds*, Invent. Math.**82**(1985), no. 2, 307–347. MR**809718**, DOI 10.1007/BF01388806 - Mikhael Gromov,
*Soft and hard symplectic geometry*, ICM Series, American Mathematical Society, Providence, RI, 1988. A plenary address presented at the International Congress of Mathematicians held in Berkeley, California, August 1986; Introduced by Jeff Cheeger. MR**1055581** - Phillip Griffiths and Joseph Harris,
*Principles of algebraic geometry*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR**507725** - V. Guillemin and S. Sternberg,
*Birational equivalence in the symplectic category*, Invent. Math.**97**(1989), no. 3, 485–522. MR**1005004**, DOI 10.1007/BF01388888 *Séminaire Bourbaki, 13ième année: 1960/61. Textes des conférences. Exposés 205 à 222*, Secrétariat mathématique, Paris, 1961 (French). Deuxième édition, corrigée. 3 fascicules. MR**0151353**- Kenneth Intriligator,
*Fusion residues*, Modern Phys. Lett. A**6**(1991), no. 38, 3543–3556. MR**1138873**, DOI 10.1142/S0217732391004097 - Frances Kirwan,
*On the homology of compactifications of moduli spaces of vector bundles over a Riemann surface*, Proc. London Math. Soc. (3)**53**(1986), no. 2, 237–266. MR**850220**, DOI 10.1112/plms/s3-53.2.237 - Shoshichi Kobayashi,
*Differential geometry of complex vector bundles*, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987. Kanô Memorial Lectures, 5. MR**909698**, DOI 10.1515/9781400858682 - P. E. Newstead,
*Introduction to moduli problems and orbit spaces*, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 51, Tata Institute of Fundamental Research, Bombay; Narosa Publishing House, New Delhi, 1978. MR**546290** - Ruan, Y.,
*Toplogical sigma model and Donaldson type invariants in Gromov theory*, preprint, 1993. - J. Sacks and K. Uhlenbeck,
*The existence of minimal immersions of $2$-spheres*, Ann. of Math. (2)**113**(1981), no. 1, 1–24. MR**604040**, DOI 10.2307/1971131 - Stein Arild Strømme,
*On parametrized rational curves in Grassmann varieties*, Space curves (Rocca di Papa, 1985) Lecture Notes in Math., vol. 1266, Springer, Berlin, 1987, pp. 251–272. MR**908717**, DOI 10.1007/BFb0078187 - Michael Thaddeus,
*Conformal field theory and the cohomology of the moduli space of stable bundles*, J. Differential Geom.**35**(1992), no. 1, 131–149. MR**1152228** - Michael Thaddeus,
*Stable pairs, linear systems and the Verlinde formula*, Invent. Math.**117**(1994), no. 2, 317–353. MR**1273268**, DOI 10.1007/BF01232244 - Tiwari, S., preprint.
- Cumrun Vafa,
*Topological mirrors and quantum rings*, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 96–119. MR**1191421** - J. G. Wolfson,
*Gromov’s compactness of pseudo-holomorphic curves and symplectic geometry*, J. Differential Geom.**28**(1988), no. 3, 383–405. MR**965221**, DOI 10.4310/jdg/1214442470 - Edward Witten,
*Topological sigma models*, Comm. Math. Phys.**118**(1988), no. 3, 411–449. MR**958805**, DOI 10.1007/BF01466725 - Zagier, D., unpublished.

## Additional Information

**Aaron Bertram**- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
- MR Author ID: 246391
- Email: bertram@math.utah.edu
**Georgios Daskalopoulos**- Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
- MR Author ID: 313609
- Email: daskal@gauss.math.brown.edu
**Richard Wentworth**- Affiliation: Department of Mathematics, University of California, Irvine, California 92717
- Email: raw@math.uci.edu
- Received by editor(s): June 8, 1993
- Received by editor(s) in revised form: November 22, 1994, and March 2, 1995
- Additional Notes: The first author was supported in part by NSF Grant DMS-9218215.

The second author was supported in part by NSF Grant DMS-9303494.

The third author was supported in part by NSF Mathematics Postdoctoral Fellowship DMS-9007255. - © Copyright 1996 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**9**(1996), 529-571 - MSC (1991): Primary 14C17; Secondary 14D20, 32G13
- DOI: https://doi.org/10.1090/S0894-0347-96-00190-7
- MathSciNet review: 1320154

Dedicated: Dedicated to Professor Raoul Bott on the occasion of his 70th birthday